An ideal gas can be expanded form an initial state to a certain volume through two different processes `PV^(2) =` constant and (ii) `P = KV^(2)` where `K` is a positive constant. Then

Options

A. Final temperature in (i) will be greater than in (ii)
B. Final temperature in (ii) will be greater than (i)
C. Total heat given to the gas in (i) case is greater than in (ii)`
D. Total heat given to the gas in (ii) case is greater than in (i)

Explanation

To solve the problem step by step, we will analyze the two processes given for the expansion of an ideal gas and determine the relationship between the final temperatures and the total heat given to the gas in each case.

### Step 1: Analyze the first process \( PV^2 = \text{constant} \)

1. Start with the equation for the first process:
\[ PV^2 = K \quad \text{(where K is a constant)} \]

2. From the ideal gas equation, we know:
\[ PV = nRT \]
Rearranging gives:
\[ P = \frac{nRT}{V} \]

3. Substitute \( P \) into the first process equation:
\[ \left(\frac{nRT}{V}\right)V^2 = K \]
Simplifying this, we get:
\[ nRTV = K \]

4. Rearranging gives:
\[ TV = \frac{K}{nR} \quad \text{(constant)} \]
This implies:
\[ T \propto \frac{1}{V} \quad \text{(Temperature is inversely proportional to Volume)} \]

5. Since the gas is expanding (volume is increasing), the temperature \( T \) will decrease.

### Step 2: Analyze the second process \( P = KV^2 \)

1. Start with the equation for the second process:
\[ P = KV^2 \]

2. Again, using the ideal gas equation:
\[ PV = nRT \quad \Rightarrow \quad P = \frac{nRT}{V} \]

3. Substitute \( P \) into the second process equation:
\[ \frac{nRT}{V} = KV^2 \]
Rearranging gives:
\[ nRT = KV^3 \]

4. Rearranging gives:
\[ T = \frac{KV^3}{nR} \quad \text{(Temperature is directly proportional to the cube of Volume)} \]

5. Since the gas is expanding (volume is increasing), the temperature \( T \) will increase.

### Step 3: Compare the final temperatures

From the analysis:
- In the first process, as the volume increases, the temperature decreases.
- In the second process, as the volume increases, the temperature increases.

Thus, we conclude:
\[ T_2 > T_1 \quad \text{(Final temperature in the second process is greater than in the first)} \]

### Step 4: Total heat given to the gas

1. The total heat \( Q \) given to the gas can be related to the change in temperature and the specific heat capacity.
2. Since the temperature in the second process is higher than in the first, it can be inferred that the total heat given to the gas in the second process is greater than in the first process.

### Final Conclusion

- The final temperature in the second process is greater than in the first process.
- The total heat given to the gas in the second process is greater than in the first process.